Cambridge Additional Mathematics

Integration (Chapter 15) 417

EXERCISE 15B

1aFind the antiderivative of:

i x ii x^2 iii x^5 iv x¡^2 v x¡^4 vi x

1

(^3) vii x¡
1
2
b Predict a general rule for the antiderivative of xn, for n 6 =¡ 1.
2aFind the antiderivative of:
i e^2 x ii e^5 x iii e
1
2 x iv e^0 :^01 x v e¼x vi e
x
3
b Predict a general rule for the antiderivative of ekx wherekis a constant, k 6 =0.
3 Find the antiderivative of:
a 6 x^2 +4x by first differentiating x^3 +x^2 b e^3 x+1 by first differentiating e^3 x+1
c
p
x by first differentiating x
p
x d (2x+1)^3 by first differentiating (2x+1)^4.
Sir Isaac NewtonandGottfried Wilhelm Leibnizshowed the link between differential calculus and the
definite integral or limit of an area sum we saw inSection A. This link is called thefundamental theorem
of calculus. The beauty of this theorem is that it enables us to evaluate complicated summations.
We have already observed that:
If f(x) is a continuous positive function on an interval
a 6 x 6 b then the area under the curve between x=a
and x=b is
Zb
a
f(x)dx.

Discovery The area function

#endboxedheading

Consider the constant function f(x)=5.

We wish to find anarea functionwhich will give the area

under the function between x=a and some other value of

xwhich we will callt.

The area function is A(t)=

Zt

a

5 dx

=shaded area in graph

=(t¡a)5

=5t¡ 5 a

) we can write A(t) in the form F(t)¡F(a) where F(t)=5t or equivalently F(x)=5x

C The fundamental theorem of calculus

y

x

y = f(x)

O ab

a t x

y

y=5

t-a

O

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_15\417CamAdd_15.cdr Monday, 7 April 2014 3:58:07 PM BRIAN